We study the problem of assigning objects. There may be multiple copies of each object. Each agent is assigned at most one. Monetary transfer is not allowed. We require a rule to be fair and efficient. We introduce an axiom that is a natural weakening of no-envy. We call it “bounded no-envy”. It states that for each pair of agents who receive objects of the same rank in their respective preferences, each of them should find his assignment at least as desirable as the other’s. Bounded no-envy is compatible with Pareto efficiency. In particular, the immediate acceptance rules (a.k.a. the “Boston mechanisms”) satisfy both. Our main result is that the immediate acceptance rules are the only rules satisfying weak non-wastefulness, bounded no-envy, rank-respecting invariance, resource monotonicity, and bilateral consistency. •We study the problem of assigning objects.•We introduce a fairness axiom called “bounded no-envy”.•Bounded no-envy is compatible with Pareto efficiency.•We study the logical relations between bounded no-envy and favoring higher ranks.•Our main result is a characterization of the immediate acceptance rules.